The size of 2-Selmer groups for the \fracπ3-congruent number problem

2026/02/28 by Kushal Bhowmick, Aprameyo Pal
#math.NT

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Abstract

Our main objective in this paper is to study the average rank of the 2-Selmer group of the elliptic curve associated with the \fracπ3-congruent number problem. Following Heath-Brown's strategy, we could find an asymptotic formula for the size of the relaxed 2-Selmer groups, which has several consequences towards the average of 2-Selmer ranks and \fracπ3-congruent number problem. Indeed, we could find an unconditional positive density of 2-Selmer rank being 1 or 3, among the positive square-free integers n≡ 13\pmod24 having all the prime divisors congruent to 1 modulo 4 and an unconditional positive density of 2-Selmer rank being 0 or 2, among the positive square-free integers n≡ 5\pmod24 having all the prime divisors congruent to 1 modulo 4.

Citations

• Quadratic twists of tiling number elliptic curves
• Genus Periods, Genus Points and Congruent Number Problem
• Congruent Numbers and Heegner Points

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